Let R be commutative ring with identity and let M be any unitary left R-module. In this paper we study the properties of ec-closed submodules, ECS- modules and the relation between ECS-modules and other kinds of modules. Also, we study the direct sum of ECS-modules.

 Let R be a commutative ring with unity, let M be a left R-module. In this paper we introduce the concept small monoform module as a generalization of monoform module. A module M is called small monoform if for each non zero submodule N of M and for each   f ∈ Hom(N,M), f ≠ 0 implies ker f is small submodule in N. We give the fundamental properties of small monoform modules. Also we present some relationships between small monoform modules and some related modules

   In this paper ,we introduce a concept of Max– module as follows: M is called a Max- module if ann N R is a maximal ideal of R, for each non– zero submodule N of M;       In other words, M is a Max– module iff (0) is a *- submodule, where  a proper submodule N of M is called a *- submodule if [ ] : N K R is a maximal ideal of R, for each submodule K contains N properly.       In this paper, some properties and characterizations of max– modules and  *- submodules are given. Also, various basic results a bout Max– modules are considered. Moreover, some relations between max- modules and other types of modules are considered.

  In this paper we introduce the notion of semiprime fuzzy module as a generalization of semiprime module. We investigate several characterizations and properties of this concept.

     The new type of paranormal operators that have been defined in this study on the Hilbert space, is  paranormal operators. In this paper we introduce and discuss some properties of this concept such as: the sum and product of two paranormal, the power of paranormal. Further, the relationships between the paranormal operators and other kinds of paranormal operators have been studied.

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

      Let R be a commutative ring with unity. And let E be a unitary R-module. This paper introduces the notion of 2-prime submodules as a generalized concept of 2-prime ideal, where proper submodule H of module F over a ring R is said to be 2-prime if , for r R and x F implies that  or . we prove many properties for this kind of submodules, Let H is a submodule of module F over a ring R then H is a 2-prime submodule if and only if [N ] is a 2-prime submodule of E, where r R. Also, we prove that if F is a non-zero multiplication module, then [K: F] [H: F] for every submodule k of F such that H K. Furthermore, we will study the basic properties of this kind of submodules.

   Let R be a ring with identity and Ą a left R-module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R-module A is s-compressible if for any nonzero submodule B of A there exists a small f in Hom_R(A, B). An R-module A is s-prime if for any submodule B of A, ann_R (B) A is small in A. These concepts and related concepts are studied in as well as  many results consist properties and characterizations are obtained.  

Let R be a ring with 1 and W is a left Module over R. A Submodule D of an R-Module W is small in W(D ≪ W) if whenever a Submodule V of W s.t W = D + V then V = W. A proper Submodule Y of an R-Module W is semismall in W(Y ≪_S W) if Y = 0 or Y/F ≪ W/F ∀ nonzero Submodules F of Y. A Submodule U of an R-Module E is essentially semismall(U ≪es E), if for every non zero semismall Submodule V of E, V∩U ≠ 0. An R-Module E is essentially semismall quasi-Dedekind(ESSQD) if Hom(E/W, E) = 0 ∀ W ≪es E. A ring R is ESSQD if R is an ESSQD R-Module. An R-Module E is a scalar R-Module if, ∀ , ∃ s.t V(e) = ze ∀ . In this paper, we study the relationship between ESSQD Modules with scalar and multiplication Modules. We show that

    The problem statement discussed in this paper is a new technique for the presentation of painterly rendering that uses a K-mean segmentation to divide the input image into a set of regions (depending on the grayscale of the regions). Segmenting the input image helps users use different brush strokes and easily change the strokes' shape, size, or orientation for different regions. Every region is painted using different brush kinds. The properties of the brush strokes are chosen depending on the region's details. The brush stroke properties, such as size, color, shape, location, and orientation, are extracted from the source image using statistical tools. The number of regions is set up manually and depends on the input image. This